General expanded form of (x+y+z)^k

[sludger13]

Hi,
(hope it doesn't seem so weird),
I'm looking for a general expanded form of
$(x+y+z)^{k}$, $k\in N$

$k=1$:
$x+y+z$

$k=2$:
$x^{2}+y^{2}+z^{2}+2xy+2xz+2yz$

$k=3$:
$x^{3}+y^{3}+z^{3}+3xy^{2}+3xz^{2}+3yz^{2}+3x^{2}y+3x^{2}z+3y^{2}z+6xyz$

$k=4$:
$x^{4}+y^{4}+z^{4}+4xy^{3}+4x^{3}y+4xz^{3}+4x^{3}z+4yz^{3}$
$+4y^{3}z+6x^{2}y^{2}+6y^{2}z^{2}+6x^{2}z^{2}+12x^{2}yz+12xy^{2}z+12xyz^{2}$

The elements are obviously determined by combinations of their powers, which sum is always $k$.
I just cannot find the algorithm for element's constants.

 

[MisterX]

http://mathworld.wolfram.com/MultinomialSeries.html
http://en.wikipedia.org/wiki/Multinomial_theorem

 

[sludger13]

Thanks, I completely forgot to check out the factorials :)